If I have a quasi experimental design, pre-test, post-test, I thought the ANCOVA needed to be used with a non equivalent group? Then do you need to adjust for a regression/measurement error?
Someone told me I could use ANOVA but I thought you couldnt unless you have random assignment?

You can use an ANOVA..it doesnt have to random assignment. If it is pre-test and post-test however, you have to do paired samples t-tests