After a dominant display at Prenton Park, Pools conceded in the 89th minute as a Josh Hawkes corner was headed in by Tom Davies at the back post to secure a 1-0 win for the hosts.
Challinor admitted he took his frustrations out on a dressing room bin following the smash and grab from his former club.
“I’m massively disappointed and frustrated,” he said. “I’m not good at controlling my emotions and focusing on the positives after losing a game or losing anything really.
“I'd have taken a point as the game progressed but we didn’t want it to be a hard luck story because of how we were first half and losing the game.
“We deserved to win the game and it’s easy for me as a manager to say that but I don’t think anybody here who watched the game could have any qualms that the better team lost today.
“That’s football sometimes and we’re guilty of not taking our opportunities in front of goal but not through lack of trying.
“You can look at it first half and be critical of the fact that we had lots of situations where we put great balls into the box but their goalkeeper hasn't had a save to make.
"If you’re standing in the middle of the six yard box, you can’t fail to have an opportunity to score from that.
"We have to put them away because it changes the dynamic of the game and we paid the ultimate price later on.
"A ‘Luke Armstrong’ would have scored three today, I know players have moved on but we've got to learn from that and not make it a hard luck story where we've lost through a lapse of concentration which is exactly what happened.
“We were outstanding today but I’m not pleased at all, we’ve lost.
"If we play great all season but go down then of course I won't be happy. We didn't have luck on our side and we were brilliant today but it's about winning games.”