Yes, I can think of sentences that use "either" that are positive. You should check either that one or the other one. I should bring either a hat or an umbrella because it is going to rain. Either my mom or dad will drive me to school.

No, you can't say that. If it's a positive sentence, you need a positive form of the adverb meaning 'in addition': You should check that one, too. You should check that one as well. You should also check that one. You would only use 'either' in this context if the sentence is negative: You shouldn't check that one, either.

Common use [You shouldn't check that one either]. // [ You should check that one too.] Affirmative use with 2 options/choices: [We can either travel now or wait until everything's done.]