This page will help you download Java on school computers without admin. This could be used for many things, including minecraft.
STEP 1: Download Java 64-Bit here.
STEP 2: Extract the files.
BOOM. You're done.
In every launcher, there should be a excutable java, or java path. Click "browse" then go to the java you downloaded, and then find the bin folder. Inside that, you scroll down until you see "javaw.exe". Open that, and you're done.
Note: There should also be JVM Arguments. Use this command (Copy all of it): -Xmx6G -XX:+UnlockExperimentalVMOptions -XX:+UseG1GC -XX:G1NewSizePercent=20 -XX:G1ReservePercent=20 -XX:MaxGCPauseMillis=50 -XX:G1HeapRegionSize=32M