While he is portrayed as eccentric and prone to strange behavior, there is no evidence to suggest he has any sort of mind control abilities. His interactions with Alice are more a result of his own eccentricities and the strange, surreal world they are in.
It's important to remember that the Mad Hatter, like many of the characters in Wonderland, is a whimsical, fantastical creation. His actions are meant to be absurd and nonsensical, rather than indicative of any supernatural abilities.
The idea of the Mad Hatter having mind control might come from the fact that he sometimes seems to be manipulating or confusing Alice, but this is more likely due to his unpredictable nature and the surreal setting of Wonderland.