Is there any regulation on how often a private aircraft should be washed?
Yes
...sort of.
There is a general requirement to clean privately operated aircraft as part of the annual inspection. There is no requirement to actually accomplish an annual inspection for a non-flying aircraft stored in a hanger. However, for an aircraft to be airworthy, the mechanic performing the annual inspection shall accomplish the required cleaning as part of the inspection. See 14 CFR 43, Appendix D:

(a) Each person performing an annual or 100-hour inspection shall, before that inspection, remove or open all necessary inspection plates, access doors, fairing, and cowling. He shall thoroughly clean the aircraft and aircraft engine.

Note that the annual inspection is generally the only inspection required of many privately operated aircraft. For aircraft operated commercially for flight instruction or carrying passengers for hire, the 100 hr inspection is also required, which might increase the required cleaning interval.
This regulation applies to all aircraft subject to FAA regulation. There are otherwise no special FAA regulations regarding aircraft washing or cleaning that are applicable to the Great Lakes area.

There may also be manufacturer recommended wash intervals. However, these are not regulations.
Operating certain types of aircraft under an approved maintenance program, the manufacturer could conceivably mandate a wash schedule which would then become mandatory in order to maintain continued airworthiness under that approved maintenance program. Such a mandate could conceivably apply to a specific geographic area based on climate issues. However, in the context of private aircraft, the requirement for such maintenance programs would only apply to certain large, turbojet, or multi-engine turbine aircraft. For this reason, neither of the types originally tagged in your question (Cessna 172 and 208) would be subject to a requirement for an approved maintenance program, though the Citation series originally mentioned in your title would be.
Moreover, if a manufacturer instituted such a change to their maintenance programs, such a change would generally only affect newly approved maintenance programs. Existing maintenance program approvals are &quot;frozen&quot; meaning new manufacturer maintenance program recommendations are recommendations only.

There may be local ordinances that limit water usage or otherwise regulate activities involved in aircraft washing.
Questions regarding EPA regulation are beyond the scope of this site.