How very unprofessional. It takes a team of at least 4 qualified professionals in different health/education roles, to make any kind of justified diagnosis. They have no right to throw that kind of pressure onto a parent. Although many traits that you have mentioned are indeed popularly considered to be autistic traits, the vast majority of ASD students I have worked with have no issues with making eye contact. Routine isn't synonymous with Autism, many children at 6 years old like routines.