I think everyone here knows it's not true, but you're probably not going to convince her that the government experts, university researchers, talk show hosts, late night comedians, magazine writers, TV people, and her doctor are all wrong about the same thing in the same way. You're one voice up against a multitude that shouts the conventional wisdom every day. Until a person takes the red pill by her own choice, arguing against all that just makes you look like a crackpot.
So you could try another tack: assume what she believes is true, but baffle her with statistics. According to the Interwebs, the chance of a man developing colon cancer in his lifetime is about 5%. That's the average, so if you assume that people with bad diets and good diets are spread fairly evenly on either side of that number, and a bacon-and-steak diet gives you 5 times the chance that a fish-chicken-veggies diet does, that would mean the best diet would give you a chance of 1.65%, and the worst would give you 8.25%. (Yes, I'm shamelessly oversimplifying the math, and it would depend on the standard deviation and the shape of the curve, etc. This is just back-of-the-napkin stuff to make a point.)
So, even if that worst-case scenario were true, would it be worth basically giving up red meat for the rest of your life except for a treat a few times a year, to reduce your chance of colon cancer from 1-in-12 to 1-in-60? Also, about 2/3 of colon cancer patients survive beyond the 5-year point, so if you look at just fatalities, it's more like the difference between .55% and 2.75%, or 1-in-182 versus 1-in-36. Also, the median age for developing colon cancer is 70, which means half of the cases happen beyond that age, near the average life expectancy anyway, and very few cases occur before age 50. Even if a meatless diet protected against it, you'll have time to adjust later. (I'm assuming here that you're not near 50, since you need to justify your eating habits to your mother.)
Presumably, you'd also have to adopt the rest of the typical red-meatless man's lifestyle, since we don't know how much of this is correlation with other aspects of that. So you'd not only have to give up meat and other "bad" foods like butter and salt, but you'd also have to start wearing cardigans, watch a lot of Alan Alda movies, get one of those "This is what a feminist looks like" t-shirts, and call your pet dog or cat your "companion animal." I'm just not sure the extra years would be worth it.
Like most health scares (and other scares), even if it's true, it's not that significant. It's the difference between a small chance and a somewhat smaller chance. If you went through life dodging everything that had a 1-in-36 chance of hurting you someday, you'd be paralyzed.