If a slave had converted prior to being acquired, do they also get released on Shmita and the Jubilee year?
A convert cannot be bought (Rambam, Avadim 1:2), so this question is invalid.
A convert may not sell himself as a servant. This is derived from Leviticus 25:41: "And he shall return to his family" - i.e., it is speaking about someone who has a family within the Jewish faith.
(Edited) A non-Jew who converts normally (not a גרות לשם עבדות, a conversion for the sake of slavery) cannot be sold as an עבד עברי, as pointed out in this answer. He can neither be sold for theft, nor can he sell himself out of poverty. The question is thus moot.
(The differences between a גר צדק and a born Jew are minute. The only other difference that comes to mind is the limitation on serving on a Beis Din (that adjudicates capital cases) and some similar שררא restrictions, like being appointed king.)
There is a special form of conversion that is required for עבדים כנענים, when a non-Jew is acquired as a slave, whereupon he becomes a Canaanite slave, and remains with his new master until explicitly freed. The exact laws are discussed by the Rambam in Laws of Slaves (Avadim 8:12, for one location). This is the normal way Canaanite slaves become that way.