Even though it's often used to address multiple people, "anyone" is a singular noun, which refers to each individual person in a group separately – literally any one of the people in the group – unlike the collective "everyone".
So since "anyone" is singular, the prototypically correct verb choice in your sentence is "he is" (or "she is").
If anyone cares to join me in this campaign, either now or in future, he is most welcome.
However, picking a pronoun (either "he" or "she") in such a case can be seen as exclusionary if not overtly sexist, especially in contexts where the group of people that makes up "anyone" is actually comprised of people of more than one gender.
That leaves us with the awkward "he/she" construction:
If anyone cares to join me in this campaign, either now or in future, he or she is most welcome.
To avoid this problem, especially in spoken English, most native speakers will opt for singular "they" in this context instead.
If anyone cares to join me in this campaign, either now or in future, they are most welcome.
Use of singular "they" is picking up in written English as well, though.
This year (2017), the AP Stylebook – the authoritative source for journalistic style in the United States, infamous for being especially slow to update their style rules to reflect changing usages – finally put out guidance on limited use cases where even the AP finds singular "they" appropriate.
As for the question about "whether" versus "either," I don't have a strong opinion on that. As a native speaker, my instinct is that they are more or less the same.
"Whether" might come with a tinge more uncertainty attached to it than "either" – so maybe "either" is more optimistic that someone might actually join the campaign? – but honestly that's a very subtle distinction, and I wouldn't bat an eyelash either way. Both options are perfectly acceptable to me.