Why must there be degeneracy in the orbitals to have a Jahn-Teller effect?
In the abstract to the original Jahn-Teller paper it is stated:[1]


  We shall show that stability and degeneracy are not possible simultaneously unless the molecule is a linear one.


Thus the Jahn-Teller effect is specifically about the removal of a degeneracy, and that that will always lower the total energy of the system. Thus you can not have a Jahn-Teller effect without a degeneracy in the reference, fully symmetric state.



Reference:
[1]: H. A. Jahn, E. Teller, Proc. Royal Soc. A 1937, 161, 220–235. DOI: 10.1098/rspa.1937.0142.