Eykyn takes over the high profile director of communications role from Paul Fincham, who left the utilities giant in October this year without a job to go to.
He will report to British Gas director of corporate affairs Christine McGourty and is expected to join British Gas at the beginning of February.
Eykyn, who has been in his current position for five years, said: "I've loved my time in government during what's been a fascinating period of huge change for everyone in the comms profession.
"It'll be a wrench to leave the superb team I led at the Department for Communities and Local Government. "But I'm thrilled to be joining Christine’s team at British Gas next year."
Before his position at the DCLG Eykyn was a senior communications consultant at Pricewaterhouse Coopers, a role he held for four years. He is also a former journalist for BBC News.
Alex Aiken, executive director of government communications, said: "George’s appointment is a gain for British Gas, but a loss to the Department of Communities and Local Government. He is a first class communicator and is highly respected around Whitehall."