I’m a Southerner, and I’m willing to bet that the lighter reads “Forget, Hell!” (note the punctuation) rather than the much more uncanny “Forget Hell.” Although I’ve heard/seen this phrase a handful of times, I could scarcely find anything on its origins—it appears most frequently on novelty plates or lighters like the one you described. That said, the meaning becomes a little more clear with the punctuation added; it should read more like “Forget? Hell!” in conveying the soldier’s chagrin at the notion he forget the violence and animosity of the Civil War. Contrite Southerners regularly referred to the war as the “Late Unpleasantness” in an effort to distance themselves from the cause or rather the humiliating defeat, whereas zealous confederate sympathizers can still be heard parroting phrases such as “The South will rise again!” Considering the decor of the lighter, you can assume it holds a candle to the latter. You might see Maxwell, Shields, and Whayne’s “The Ongoing Burden of Southern History: Politics and Identity in the Twenty-First Century South” (2012) for more on how writers and politicians adapted the “Lost Cause” myth during and after Reconstruction.

It's impossible to say for sure what the maker of that image had in mind when they made it. But I think the most likely reference being made is to a famous quote attributed to Union general William Tecumseh Sherman, "War is Hell." Sherman was one of the Union's most able generals, remembered for his "march to the sea" in 1864. His army marched 300 miles from Atlanta to Savannah, destroying everything they didn't seize and use themselves, with the goal of destroying the South's ability to wage war. The full context of the quote is a speech supposedly given to graduates of the Michigan Military Academy several years after the war:

I’ve been where you are now and I know just how you feel. It’s entirely natural that there should beat in the breast of every one of you a hope and desire that some day you can use the skill you have acquired here. Suppress it! You don’t know the horrible aspects of war. I’ve been through two wars and I know. I’ve seen cities and homes in ashes. I’ve seen thousands of men lying on the ground, their dead faces looking up at the skies. I tell you, war is Hell! Source

Now, the exact wording, and even the actual historicity, of the quote are both unclear. But the overall sentiment matches Sherman's other writings and statements, and the phrase "War is Hell" has become well known.

Again, it's impossible to know exactly what the maker had in mind. But that sentiment beside the Confederate flag could be meant to imply that the South shouldn't fear the consequences of war, with the further implication that "the South's gonna do it again."

If that is, if fact, the intended message, then I would direct anyone who believes it to another quote from Sherman, this one from before the war:

You people of the South don't know what you are doing. This country will be drenched in blood, and God only knows how it will end. It is all folly, madness, a crime against civilization! You people speak so lightly of war; you don't know what you're talking about. War is a terrible thing! You mistake, too, the people of the North. They are a peaceable people but an earnest people, and they will fight, too. They are not going to let this country be destroyed without a mighty effort to save it … Besides, where are your men and appliances of war to contend against them? The North can make a steam engine, locomotive, or railway car; hardly a yard of cloth or pair of shoes can you make. You are rushing into war with one of the most powerful, ingeniously mechanical, and determined people on Earth — right at your doors. You are bound to fail. Only in your spirit and determination are you prepared for war. In all else you are totally unprepared, with a bad cause to start with.

A prescient statement, if ever there was one.