There is already fairly substantial proof that Russia has directly interfered with the US election and even if the rumors about Trump are not true and not substantiated, we still need to reduce the amount of influence that Russia has on the US Goverment. They are not an ally and cannot be trusted.
I don't think Rex Tillerson would hire an inexperienced person to head the PR section of his company, so how can he claim to be a good candidate for Secretary of State? Plus his ties to the oil industry and to Russia present plenty of conflicts of interest. I can't believe the next administration is going to actively work to benefit Russia.
It would seem that Donald Trump and his administration have a strong love for Russia that seems mighty suspicious. Now I understand this isn't the Cold War anymore but Russia has been far from friendly with the US as of recently and Trump is acting like nothing is wrong. If Rex Tillerson is found to have strong ties with Russia and with the recent information about Russian hacking he should be ineligible for Secretary of State because we don't need traitors in the White House.
Knowing what we know about the Russian hacking and election interference, I think that is one good reason Tillerson not to be eligible. His Russian ties make him a liability, given the relations that we have with the Russian government. And to think, this is just one of the very valid reasons he shouldn't be eligible.