Adam Austin said bosses at the Morrisons store in Portsmouth told him he faced disciplinary action after he refused to remove the items.
The 28-year-old has now been suspended until he attends a meeting with management on Friday.
A full copy of the disciplinary letter
Mr Austin told the Portsmouth News: "I was disappointed and partly furious.
"I have got friends in the army and friends who have been in the army. What with the issue with Lee Rigby, I wore them out of respect."
A Morrisons spokesman explained that staff were generally prohibited from wearing extra items to prevent objects falling into food products
He said: "It's all about food hygiene. The rules are the same in a restaurant kitchen as they are in a supermarket food preparation area.
"We have a dress code and the dress code is primarily around the health and safety of the staff member and the members of the public."
Mr Austin with the band he wished to wear in memory of Lee Rigby
But he added the company had reviewed its guidelines to allow staff in non-food preparation areas to wear a registered charity wristband.
But how this applies to Mr Austin is unclear at present, as he is employed in a fresh food preparation area as well as at the checkouts.
The spokesman added: "We understand that it's important to many people to show support for our armed forces. Morrisons is a long time supporter of the Royal British Legion's Poppy Appeal and we welcome Help for Heroes fundraisers into stores.
"As a company we encourage store colleagues to show their support for the Royal British Legion by wearing poppies in October and November.
"We currently ask that colleagues adhere to a company dress code which precludes bracelets and pins. However, we have reviewed these guidelines and colleagues working in non-fresh food preparation areas will now be permitted to wear a registered charity wristband."
A petition has been started on Change.org to allow Mr Austin to wear his tribute to Lee Rigby and force Morrisons to make an apology, donating £20,000 to Help For Heroes.