MUST a landowner lease land to secure royalties...

No, he can remained unleased and will be entitled to 100% of the royaltys for his 2 acres after the cost of the well is paid out of production. 28 months is just an arbitrary number it could be months or years there is no set time period, it depends on how the rate of production of the well and the market price for the oil or gas. He is also probably passing up bonus money that would be paid for just signing the lease, the increased royaly money would make up for this, but the bonus money is guaranteed while future royaltys carry risk. The well may not produce as expected, the life of the well may be cut short by formation problems, or based on the production of wells that are already drilled in the area, they decide not to drill his section. The latter has happened to me twice, so am glad I took the bonus money and leased.

Its not.

Actually the operator is entitled to recover drilling costs X2.

To the OP, your dad should lease. The well might not pay out twice so he won't get anything or it could take a long time and by the time he gets in pay status prod may have significantly declined. So while he's getting 8/8ths he would have made more over time by leasing for a 1/4 plus a bonus payment.

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Actually the operator is entitled to recover drilling costs X2.

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In LA the unleased mineral owner only has to wait until the costs to drill and complete are recovered. Working Interest owners who go non consent have to wait until 300% of the costs are recovered, as the law was changed a few years back moving this up from 200%.
To the op: Some operators will "operate these interests to death", though and it would be risky to not lease. Leasing essentially takes all the risk off the table. If you choose not to lease, you could potentially cause the well not to be drilled should you have enough acreage to make the operator not want to carry you. By leasing you make money even if the well never reaches payout. By not leasing you will get statements you can't understand and will be facing an uphill battle getting any information out of most operators. Also the operator will not view you favorably and you will make an adversary out of most of them in this situation. I personally think LA needs to raise penalties for unleased mineral owners who refuse to lease higher than the current 100%. Some landowners can and are stalling development by not leasing. Arkansas can be as high as a 500% risk premium and wells get drilled regardless. With your 2 acres that's not going to make you a tremendous amount of money either way so you might as well lease and remove the risk.

Tell him to take the 25% and the bonus. If he refuses to sign he will take a 300% penalty, meaning the well has to pay out 3x before he sees his first royalty check. I have 80 acres in the Haynesville and I leased for $200 an acre and 1/4 and am very happy.

plus it's 2 acres, not getting rich either way he does it