What are the ways to test for protein in the canine urine?
After some research, I could primarily find two tests for testing the protein content in dog urine.

1) The Urinalysis dipstick test (reference) which has a chance of giving false positive results (reference). Here is the link to a lab test done using the dipstick method and the results that the researchers got. 



A sample Dipstick (reference)

2) The second test would be the benzethonium chloride test (reference).


  This test precipitates the protein and increases the turbidity of the
  sample. The turbidity is proportional to the protein concentration.
  This technique is very sensitive and can yield accurate results in
  samples with very low protein concentrations (&lt; 20 mg/dL).


A lab study done using this procedure to precipitate protein (reference).


  The urine sample is best read within 30 minutes of collection
  otherwise certain factors (like crystals, bacteria and cells) may
  alter their composition (dissolve or multiply) over time. If you
  collect a urine sample at home, please drop it off at your veterinary
  clinic as soon as possible (reference).


Reasons for excess protein if present in the urine of your pet is listed in this article.