Before we get into axle breakage, let's discuss your axle spacing, which probably isn't going to work going from a 6spd to an 8spd cassette unless you install a new, longer axle with more spacers, and you'll need to re-dish the wheel slightly.
In terms of breakage, I've seen all speeds of freewheel axles break over time. Freewheel hubs' axles are just more likely to break because of the leverage you can get over the axle. Here's a great diagram that illustrates this point:
The red circles are where the bearings brace against the axle. The drive side of a freewheel hub has a lot more axle acting as a lever than does a cassette hub, and this is the primary reason they break.
Is an 8spd spaced freewheel hub more likely to break than a 6spd spaced freewheel hub? Yes it is, but marginally. Even working in a shop for years, I cant say that I saw a statistically significant sample of broken axles come through to say that 8 speeds broke more often than 6 speeds- probably the opposite as the 6 speed hubs were older and had seen more use. The bigger comparison here is freewheel vs cassette wheels, and I can tell you without a shadow of a doubt that cassette hubs hold up better.
Since a cassette hub isn't what you're looking to use though, I would say don't concern yourself with the increased likelihood of axle breakage when going from 6spd to 8spd. You're probably going to be less likely to break the axle, assuming you buy a new axle for the conversion as you'll be starting over again on the fatigue life for the axle.