Urinalysis sticks are a quick, easy way to check for protein, sugar, ketones, blood, bilirubin and leucocytes, amongst other things. The little squares change colour to indicate the presence of these things, and how much is present - they are simple and easy to use, and doing this sort of urinalysis is something I was doing routinely on my first ward, within three months of starting nurse training.
If I recall correctly, the presence of protein and leucocytes would suggest an infection - and the nurse must have recorded the results of the analysis - if she recorded NAD (nothing abnormal discovered), then there was nothing to indicate a need for further testing.
If I were you, I would ask to see the record of that visit. If the test of your ds's urine was clear, then I would doubt there was an infection present, and the doctor was wrong to tell you that he knows there was. But if there were leucocytes (white blood cells), and/or protein, and/or blood, then she should have investigated further.