Honda are supplying engines to two teams for the first time since their return to Formula 1, supplying both Red Bull and Toro Rosso who in total completed nearly 1,000 laps between them, proving the reliability of the engine.
Egginton said (as quoted by Autosport): "The engine programme has been fantastic.
"Honda have been working along in the background doing their thing. It's fantastic.
"They've been causing us no compromise, it's been great, the PU situation's been fantastic. We've sort of picked up where we left off at the end of last year.
"Plus we're familiar with their guys now, it's quite seamless for us, the majority of the guys in the garage are the same.
"So, from the Honda side, they're just getting on with it and we're leaving them to run their programme."
Honda endured a tough few seasons of engine and reliability problems with McLaren before providing for Toro Ross throughout last year.
The team performed well and the Japanese manufacturers proved enough to convince Red Bull to switch from Renault as providers to go for the Honda engine.
After a rebuilding year with Toro Rosso in 2018 after switching from supplying McLaren, Honda is back in the spotlight this season following Red Bull's decision to move on from using Renault power.
Egginton said: "Honda were strong in development last year, they found a lot of kilowatts, [and] we moved forward very rapidly.
"Like every engine manufacturer Honda have been super busy over the winter, working on the next steps, and we're carefully validating what they've learned over the winter.
"We're introducing, step-by-step, these developments and pushing forward.
"Every team is nowhere near the edge of the operating window on the engine.
"Since this time last year the engine has taken a fantastic step forward.
"It was clear towards the end of last year that they'd made that step and they're just carrying on, so we're very happy in that respect."