I don't recall Lincoln and his government putting a draft requirement upon California for soldiers. While there were some Union units from California they were volunteers. Being born in England should have kept Charles E. Boles from the draft if he objected but there is no indication he ever was drafted.
I looked for a pension record for Charles E. Boles and could not find one within the 116th Illinois.
The soldier Charles E. Boles had been wounded at Vicksburg and he may have been a 1st Lieutenant near the end of the war. If he was then hopefully a place of birth might be listed somewhere on his CMSR's at NARA.
The 1870 United States census on ancestry listed him as living in Deer Lodge County in the Montana Territory and the nearest Post Office was reported as Silver Bow and listed his occupation as a miner with a year of birth of about 1833 and noted he was born in England.
The Montana Territory came into existence in 1864 and prior to that had been part of the Idaho Territory and Deer Lodge County did have small deposits of gold that were mined prior to the 1870's. Montana became a State in 1889.
Charles E. Boles was living with another miner named Henry Roberts from Missouri. The majority of the people living in Deer Lodge County in 1870 were from Canada or Scotland and England from looking very quickly.
I don't know how historians have connected Charles E. Boles with the soldier from the 116th Illinois? That might provide the missing link? If I see where the soldier in the 116th Illinois had been born in England as an example on his CMSR's then I would be compelled to believe this would be the same Charles E. Boles.
But for now in my opinion your theory must be considered because of the circumstantial evidence you have provided which should not be ignored.