Can a non-veg guy who eats mutton and chicken daily worship Lord Hanuman?
It is known to all followers of Sanatan Dharma that Lord Hanuman is a Vaishnava (eternal servitor or servant or a devotee of Rama or Vishnu). To worship a Vaishnava does not fit one to eat non-veg. A strict brahmachari is vegetarian by nature and again does not fit one to be non-veg.
As far as the worship is concerned, you may worship Him on fasting days or those you are not eating non-veg. But even after having eaten non-veg, you may think of Him. But do not recite the Hanuman Chalisa at any cost and it invokes Him personally, which is not right after having eaten non-veg. In many religions meat was even offered to Gods in temples.
Only later on the brahmans in India formed a committee and made it as a law not to offer meat to Gods. Later they too refused to eat meat to look superior caste than others.
About hanuman, He is considered as a helping God, and since he was a Monkey God, he was considered a vegetarian. Since we are not Veggie and neither are we Monkey gods, we can worship Hanuman even after consuming meat or fish.