Okay you're getting some things mixed up so let's go over them one at a time.
"A starship should be able to reach the speed of light and continue beyond into the superluminal realm, with constant acceleration"
This part is correct if we're looking at the performance of a gravitational field propulsion system, like the one that Miguel Alcubierre described in his 1994 paper:
[gr-qc/0009013] The warp drive: hyper-fast travel within general relativity
However there's still a heated debate about whether it's possible to travel faster than light in this manner, because the opponents have a very good argument: any field used to create the warp bubble in his model would be limited to the speed of light (even gravitational fields only propagate at the speed of light), so it's hard to imagine how any field could travel faster than its own propagation speed. But I'm not entirely convinced by this argument. If the field were contained within the material of the hull of the craft, then it might be possible for the craft to free-fall at superluminal speeds. I haven't seen an analysis of that approach in the literature yet, so for now I'm putting it in the grey basket.
"by the use of an infinite fuel source"
This part is mixing up apples and oranges: a gravitational field propulsion system doesn't require any additional energy to accelerate once the field is set up. Only rockets and other reaction propulsion systems require energy to accelerate. And in the context of special relativity, it takes an infinite amount of energy to reach C, so superluminal spaceflight is ruled out in that theory anyway.
"an addition of a magnetic shield that mimics no rest mass"
I have no idea what you're talking about here; nobody's proposed a model for a magnetic field that can mimic zero rest mass. And in fact, magnetic fields possess positive energy, so this appears to be a contradictory notion.
"thusly negating the negative effects of the speed of light barrier."
This doesn't make any sense. In special relativity, we see that photons have no rest mass, and yet they're limited to the speed of light. So we're back to the gravitational field propulsion concept, which is the only tentatively viable theoretical approach to superluminal travel. And in that model it doesn't matter what the mass of the spacecraft is; the relevant factor in this model is the dipolar gravitational field that propels it.
You shouldn't agree with that assessment because it's completely wrong. For example, take a look at the unlabeled "depart Earth...arrive planet...return Earth" diagram on that webpage:
Why Time Dilation Must Be Impossible
That's a common misunderstanding among first-year physics students. He's showing the return trip at a velocity of -C. Velocity is always positive, or zero, not negative.
So the ship is in an (positively) accelerated reference frame in both directions. The magnitude of the velocity relative to the Earth (which is always positive) and the amount of time spent at that rate of acceleration, determine the total amount of time dilation that the traveler experiences with respect to the Earth observer. So the Grandfather paradox is correct, and it's not actually a paradox after all - it's just a demonstration of the time dilation effect, which is proven beyond any doubt as DROBNJAK stated (in fact the GPS system proves both forms of time dilation all day long every day).
"short lived particles that travel a fraction of a percent of c survive for much longer than their lifetime when at rest in the lab"
That's not right. Time dilation is insignificant at a fraction of a percent of C. It takes speeds very close to C to see the really significant time dilation that we find at CERN and with muons created by cosmic rays, etc.
"But with a starship that is encased in it's own magnetic field that mimics no rest mass, the human crew and craft will not be affected by relativistic speeds, because it is safely cocooned within it's own magnetic shield and not affected by time dilation at all."
Replace the "magnetic field that mimics no rest mass" with Alcubierre's warp bubble that's defined by its positive and negative gravitational potential, and then you'll be correct: there is no time dilation with gravitational field propulsion. At least in purely theoretical terms, it might be possible to circumnavigate the entire galaxy and be back home in time for lunch, using such a system. But unfortunately nobody's come up with an even remotely viable approach to actually building such a thing, though I remain hopeful, because in my view the AAV phenomenon proves that it can be done.
