Can muscle pains in your chest cause you to be sick?

Yes, feelings such as twinges and flutters in the chest can be experienced in anyone and in most cases are harmless.

However I would get this checked out at the Doctors. You may be asked to wear an ECG monitor for 24hrs in order that they can check your heart is pumping correctly and the electrical signals to it are rhythmical. 

Most arrhythmia is benign but it is best to understand what is going on, if only to arrive at an explanation and therefore a starting point for any recommendation.

Best wishes

Earle