Speaking in his Scottish Sun column, Boyd said Celtic fans who booed their team off the pitch are wrong to think that they 'belong' in the Champions League.
He said: "Those fans who booed their team off the pitch after Tuesday’s defeat to Cluj are deluding themselves if they think they belong in the Champions League.
"It’s like they believe they’ve got a God-given right to play at that level when in actual fact they’ve got no right to think that way.
"There are already some bigger clubs than Celtic who have been eliminated. For me they’re just too arrogant, spoiled and big-headed to see it.
"They are turning up for games just expecting to see Celtic win when the reality is they’re a tiny little fish in a massive big pond when it comes to European football.
"They can kid themselves on all they like, but that’s the harsh truth. Being so dominant domestically in recent years has only papered over the cracks of their failings at European level.
"Maybe for some, losing to a very ordinary team from Romania will bring them crashing back down to earth with a bang.
"I’m not talking about the players or Neil Lennon here. I’m talking about the fans who have been living in Cloud Cuckooland for too long now."
Celtic went down 4-3 at Parkhead to Cluj having drawn the first leg 1-1, dropping down to the Europa League, where they face AIK Stockholm in the final qualifying round.
Boyd also claimed that it would benefit Rangers if Celtic make it past the Swedes and into the group stages.
"If Celtic go through then they’ll be playing their league football on a Sunday, so they’ll potentially be playing catch-up.
"That could have a huge bearing on how the season goes and shouldn’t be underestimated."