The red appearance of horse urine in snow can be normal. Horse urine can change color after being voided due to the presence of plant metabolites (pyrocatechines) in the urine that turn a red or orange color when mixed with oxygen. This happens year round, but is especially noticeable in snow. Normal horse urine should be colorless to yellow to dark yellow. If you see that the urine is red, brown, or orange as it is being voided (before it hits the snow) this can be a sign of a medical problem and you should talk with your veterinarian about it.
Source : umn.edu