In the UK, a public school is an elite private school, originally one of seven private schools given independence from direct jurisdiction by the Public Schools Act 1868: Charterhouse, Eton College, Harrow School, Rugby School, Shrewsbury School, Westminster School, and Winchester College. The list has been somewhat expanded and is now considered to cover all the fee-charging independent secondary schools that cater primarily for children aged between 11 or 13 and 18, and whose head teacher is a member of the Headmasters' and Headmistresses' Conference. Yes they generally require you to wear a school uniform (not a unifrom) and a gym kit will be part of that.
If you meant to ask about tax-payer funded schools, the answer is that it varies in Scotland. There is no official requirement although most schools do have a uniform because parents and teachers prefer it. Creates less peer group pressure and better discipline.