It sounds wrong to me, too - but seguir + gerundio makes sense, because "to continue" implies that there is some sort of action going on, and the gerund continues that action, fitting with seguir in a way that is very similar to the way that estar precedes gerunds. seguir is acting more like an auxiliary verb preceding the gerund-action.
This is very different from something such as "I like eating", where "eating" really is being treated as a noun, and thus, in Spanish, would require the use of the infinitive: Me gusta comer. In this latter exmaple, comer is the object of the stand-alone verb gustar, there is no continuous flow of action from "liking" to "eating", so the use of the gerund in Spanish would be wrong.
There is a difference between verbs and adjectives. It is the adjectives that get the o or a at the end depending on whether they describe a feminine or masculine noun.
Comer is a verb and does not take impact of gender, but instead uses a form ending in -ndo to describe something that is ongoing.
comer - seguir comiendo (to continue eating)
yo como - sigo comiendo (I continue eating)
tú comes - sigues comiendo (you continue eating)
él/ella/usted come - sigue comiendo (he/she/usted continues eating)
nosotros comemos - seguimos comiendo (we continue eating)
ellos/ellas/ustedes comen - siguen comiendo (they/ustedes continue eating)