Is my company correct in telling me I have to wait for open enrollment for short term disability and will probably be denied?

I cannot comment on whether your short term disability is subject to open enrollment or not.  You should get a copy of their policy and speak to an attorney about that part.  They should not, however, be telling you that you would not qualify unless there was some requirement that you needed to work there for a certain period of time to be eligible and you had not worked for that amount of time.  They should not be making medical decisions on whether you would be disabled or not.  That is up to the disability representative