No, convicted felons should not be allowed to carry guns. There are only three inalienable rights in the Declaration of Independence, Life, Liberty, and the Pursuit of Happiness. Gun ownership or carrying rights is not among those listed, and instead only granted in an Amendment of the Constitution. Thus, it is not an inalienable right.
While I understand that they may be a number of different felonies a person can commit. I don't think people who have abused guns or committed violent crimes with them should be allowed to carry guns. However for crimes that may not have been violent or involved guns that may be a different issue.