"Is it professional of me to play games" This alone results in a "no". However, whatever you do in your own time is up to you.
However when you use your free time at work, it might get complicated. It depends on the company. you'd need to ask yourself several questions.
- Do customers walk around the company in the hours you're playing games?
- Do other employees know of your hours? Does it matter?
- How strict is the business culture?
If customers walk around the company and see you playing games, that leaves a bad image to the company. While the customer may/may not care, your superiors absolutely will. At this point they wont care if you were on break or off work ot not. Of-course the break room is an exception.
In a more strict 9 to 5 company they will not enjoy having their employees making weird hours or doing non-working activities. In those cultures other employees will often try to copy such behaviour and possibly lower productivity.
In a less strict company, employees coudln't care less what you do during and outside of your working time. Mainly because it doesn't affect them. In such a culture you'd be able to play games whenever you want, for as long as you end up making the hours. (I doubt this is the care for your company)
As suggested before, going for the break room would be the 100% safe solution. I've only seen a few companies that have an xbox in the break-room but whenever anyone were to play on it on their break, all managers and other employees would give you weird looks. Again, it depends on the company's culture.
If you end up being unable to figure this out on your own, talk to your manager. Most managers tend to have weekly or bi-weekly 1-on-1 conversations with their employees, if you dont have one, ask for one. or simply walk by his office if he has one.
If your manager doesn't have either of these. As in no weekly conversations or personal office. Then I'd still try to catch him somewhere in private. I wouldn't feel comfortable having other employees weighing in on this topic.
Furthermore, While you might be in off-hours. You're still available at work. This in itself is quite valuable for the company. They dont need to pay you for being available, which is in their own benefit. You know, in case they need to ask you a small question.