You asked:
Is it correct to say,"I would like to have him continue(continues?) to work in the company.
Or,"I would like to have him continuing to work in the company."
The correct way to phrase this would be:
"I would like him to continue working in the company."
OR
"I would like to have him continue working in the company."
The two sentences have the same meaning exactly but the first is more concise and therefore,better.
The verb "continue" can be followed by both the gerund(verb+ing) or the infinitive to (to+verb). Using the gerund emphasises the on-going action in this case.