When did the US start refrigerating eggs on a regular basis?
The question actually brings up two separate issues: 

(1) When did the U.S. "start refrigerating eggs on a regular basis," i.e., when did the process of refrigeration become a common practice with eggs?  

Answer: late 1800s

(2) When did refrigerated eggs "become the norm," i.e., when did American consumers expect eggs to be always (or almost always) refrigerated?

Answer: trending upward significantly in the 1950s, very widely accepted for eggs in grocery stores by the 1970s, legally mandated in the 1990s

Further details on both questions below.



Preserving eggs on a mass scale has a long history in the U.S.  There's an entire chapter devoted to eggs in the book Fresh: A Perishable History (2010).  As discussed there (pp. 86-87):


  In fact, of all the perishables first touched by commercial
  refrigeration, no food proved a harder sell than the cold-stored egg. 
  Even in the fridge-friendly United States, popular distrust of the
  "storage" egg endured for decades.
  
  As refrigeration experts saw it, the basic problem was consumer
  ignorance.  People needed to give up the outdated idea that the only
  good egg was a local and recently laid egg.  But as the public saw it,
  the basic problem was uncertainty.  The egg's unrevealing nature
  became an unappealing mystery once merchants began chilling massive
  stocks.  The perceived danger lay not so much in cold storage itself
  as in the way it could be used to cheat people out of wholesome and
  fairly priced provisions.


[This section of the answer largely derives from this book.]

The first wave of egg refrigeration occurred after the mass rail transport system was developed in the mid-1800s.  By the 1860s and 1870s, eggs were sold in New York from as far away as Minnesota and Mississippi, carried there by refrigerated rail cars.  (Of course, these cars were packed with ice, rather than modern mechanical refrigeration equipment.)  By the last decades of the 1800s, eggs even began crossing between continents on refrigerated steamships: Normandy's egg prices were driven down by cheap refrigerated egg shipments from the U.S., while California received regular egg shipments from China.

But aside from egg trading, cold storage at this time was necessary because eggs were a "seasonal crop."  Hens would stop laying eggs for much of the winter, necessitating cold storage to supply eggs year-round.  While previous generations had preserved eggs temporarily in cold cellars or using various other techniques, by around 1900 the United States had a network of cold-storage food warehouses in major cities that was unparalleled elsewhere in the world.

Nevertheless, consumers had mixed feelings about "cold storage eggs."  They couldn't be sure about storage conditions, and eggs often acquired odors and flavors from other produce and food stored at these facilities.  And as the practice became more common, consumers couldn't be sure whether their eggs really were "fresh" as advertised, or whether they had spent time in storage, particularly in the cold months when few fresh eggs were laid.  Consumer guidebooks and magazines of the early 1900s suggested that cold storage eggs were a useful necessity as an ingredient, but were not of high enough quality to be eaten simply fried or scrambled.

There's a lot more to this early story -- involving consumer reactions, business advertising campaigns, political actions, governmental intervention/regulation, etc. -- but suffice it to say that cold storage eggs were both common and available (particularly in major cities) from the late 1800s onward.  But such storage was a necessity for transport and longevity, not something used yet by retail grocers and consumers.

By the early 20th century, there was already a push toward cooling fresh eggs which were not intended for long-term cold storage.  But since farm refrigeration equipment was not commonly available yet, there was little effect, and well into the 1920s there are reports of suspicious consumers who had to be convinced to try refrigerated eggs when told they were refrigerated, even though many had likely had cold storage eggs preserved in warehouses.

There was a temporary setback to the cold-storage warehouse industry with the development of techniques to make chickens lay eggs year-round.  Starting in the 1910s, a series of shifts in feeding schedules, temperature regulation, and henhouse lighting gradually led hens to lay in the "off-season."  While annual cycles didn't disappear completely until around 1970, the demand for long cold storage decreased, as consumers preferred the fresh products.



[This section of the answer is somewhat more speculative and was cobbled together from various sources.]

The true emergence of cold storage as a "norm" for the whole path from farmer to consumer happened after the rebranding of "cold-storage eggs" to "refrigerated eggs" to eliminate the stigma of eggs that were not perceived as "fresh," along with the advent of cheaper and more widespread mechanical refrigeration for farm and home use in the 1940s and 1950s.  Egg production increased significantly during WWII, and after the war, numerous studies of egg longevity convinced farmers that cooler storage conditions would reduce spoilage and thereby save them money in the long run.  By 1953, Popular Mechanics reported the modern marvel of the large-scale chicken farm, including: "Painted on the farm's big refrigerated truck is the slogan 'Eggs From Happy Hens'."

Grocers were gradually convinced of the same thing.  A report from Baltimore on retail egg practices notes that, between 1946 and 1951, the number of retail stores refrigerating eggs went from about 1/3 to 2/3 of all stores.  

Consumer recommendations for egg storage followed as well.  The cookbook You Can Cook if You Can Read (1946), for example, notes the following on "The Care of Eggs":


  Always keep your eggs in the refrigerator in a covered dish or pan. 
  They deteriorate much more rapidly in warm air than in cold.  And if
  possible, find a grocer who keeps his eggs in the refrigerator too. 
  All the grading and government inspection in the world cannot
  guarantee fresh eggs to the purchaser as long as grocers and
  householders keep them in the open air.


In the 1950s, many magazines also began running ads for refrigerators with dedicated compartments for eggs in the refrigerator.  Refrigerated trucks (as described above) were perfected in the 1950s and made refrigeration possible for all stages of transport.  Unless eggs were purchased from a small farm, by the 1960s most consumers were used to seeing eggs stored in refrigerators all the way from the farm to consumer.

Of course, the period around 1950 was also the time that egg washing became widespread, though the relationship of egg washing to refrigeration was probably more to do with a consistent production method and quality control.  (Washing methods varied greatly and were not consistently regulated for inspected eggs until the early 1970s, after a series of Salmonella outbreaks in the 1960s.)  

While it may be at least partly true that the U.S. continues to advocate refrigeration for food safety because eggs are washed (even if recent research indicates that the effect of washing on cuticle damage may be limited), the initial trend toward refrigeration occurred in an earlier period for a variety of reasons.

The holdouts among consumers who would leave eggs on the counter were probably convinced to put them in the fridge due to consistent storage among grocers and concerns about quality deterioration during temperature cycles.  (Condensation on cold eggs left out also can help bacteria migrate through the shell to contaminate the interior of the egg.)

But there was nothing actually requiring refrigeration from farm to table until the early 1990s, when more frequent Salmonella outbreaks in the 1980s convinced Congress to act.  (Regulations on refrigeration before that varied from state to state; the applicable 1963 Michigan law -- likely valid for the Detroit area -- only required egg storage temperatures below 60F.)  Since then, a number of new federal regulations have been passed, beginning with temporary refrigeration rules in the early 1990s.  The 1991 amendments to federal regulations also required all consumer egg containers to contain warnings that indicate that refrigeration is required.  These regulations -- basically requiring eggs to remain below 45F from farm to grocer -- were made permanent in 1999 and 2000, though exceptions are still available for small farms with fewer than 3,000 laying hens which pack their own eggs and sell directly to consumers.