By the 14th amendment, I presume you are referring to the Due Process clause. That clause has been interpreted fairly widely, but in general it refers primarily to legal punishments. Internal school rules have nothing to do with the law, and even in cases where they do, the law is the law, and violations of it are dealt with by a court.
As for your other question, the reasoning at the time was that "separate but equal" facilities were still equal. The court later ruled that the segregation itself was inherently harmful and unconstitutional.
The principal components of a spectrophotometer
Nursing Interventions and Prochaska's Stages of Change
Defining Climate Change
Review of Metric Prefixes
Coral growth rings as an indicator of environmental change
(left) Urine sample from an unaffected person does not change color upon exposure to air. (right) Ur