Not in elementary school or later, but in preschool it happened to me a few times.In my state, preschool is for ages 3 to 5; mine even accepted some 2-year-olds.It was, like many others, run by nuns.I remember that a few times I had my rectal temperature taken because I wasn't feeling well. I was taken to a small room that served as an infirmary, and, lying face down on the bed, my clothes were taken down and my temperature was taken.When I was 5, I remember it was summer, and I had heat stroke; I felt sick. And they took my temperature like that for the last time.I felt no shame; they often saw us undressed and took the little ones to the bathroom, and it wasn't annoying at all, but some, especially the girls, cried.
I cannot honestly ever remember having my temperature taken at school at all. Physicals and such were done at the doctor's office, and I don't remember every having to go to the nurse due to illness, but, given the times, it probably would have been done that way (I graduated 8th grade in 1976, so it seems likely, at least in the earlier grades.) It was a private parochial school, and the nurse was not full time, and a volunteer at that (She was one of the parents who would come in)