Lets face it, the party divides are too great to bridge by a quip or even a good talking point.
That great guy who always votes Republican is going to keep doing it next year and 20 years from now, no matter how persuasive your arguments. In some ways, party identification is more like Religious identification than opinion.
But that can be a good thing too. If we can convince the friendly Republicans we know of one or two important points, then they return to their friends on the other side of the great divide with new ideas, new outlooks, which will be accepted precisely because they are Republicans.
A couple key ideas and the "how to" below the fold.
First, remember the movie "A bridge too far," or at least the idea of it. Don't try to sway them to your point of view, present them with facts that shift where on the conservative ideology they fall. Or even better, give them facts that cause them, because of their values, to support your short term position.
Semi-off-topic, there are two types of compromise. In one, each side gives up "half," to meet in the middle. In the other, both sides identify what their goals actually are, discover that there are things that one side wants far more than the other, and both sides give up "one eighth," and still meet in the middle, because they gave up things they didn't care much about to get what they most wanted. Find out what the Republican actually wants, listen to them, and learn that they are not in lock step with their party or a right wing talk show host (hopefully).
Agree with their positions that you do agree with, or explain how you could agree with it. Listen, and restate, turn it from an argument to a discussion.
Now, two good points to hit them with.
"If the debt limit doesn't pass, the military will take the bulk of the cuts." It has to, of the discretionary budget, the military is more than all other spending combined. Military $740, VA $141, Homeland security $48, non-military $615 (all in billions). If we default, it will raise interest rates on the debt, by at least 1%, which means $160 billion in extra costs. Domestic spending has been cut to the bone, and when looking at how to pay that extra $160, deciding between bombers and national parks or cancer treatments will be hard for congress.
"If the debt limit doesn't pass, you could lose your job." We are right on the edge of a recession, and everyone outside the 1% feels the uncertainty. If the US defaults, not only will it hurt our government, it will seize up the bond markets, companies won't be able to borrow money, and corporate spending will plummet. This means a slowdown of every sector of the economy, and layoffs to preserve cash. Meanwhile the world economy goes into similar shocks, which means a cut to exports, so if your company sells outside the US, your sales will drop, again leading to layoffs or hour cuts.
Also, make sure to stress that the new talking point of Republicans about "Paying the interest and rolling over the principle," is illegal unless the debt limit is raised. The debt limit is "how much principle we can have," and if we roll over the principle, and then spend a single cent more, the congress has broken the law.