Did either President Lincoln, Jackson or Kennedy tried to set up any interest free monetary system?
Lincoln established interest-free currency (greenbacks):
In July 1861, Congress authorized $50,000,000 in Demand Notes. They bore no interest, but could be redeemed for specie "on demand".
As explained in History of the Last Quarter Century in the United States Scribner's Magazine volume 18, pages 71-91, (July 1895):
An act of Congress passed February 25, 1862, had authorized the issue of $150,000,000 in non-interest-bearing Treasury notes. These notes had no precedent with us since colonial times. Neither receivable for duties nor payable for interest on the public debt, they were yet legal tender for all other payments, public and private. As the Government paid its own debts with them they amounted to a forced loan.
(reference also available in book form as The History of the Last Quarter-century in the United States, 1870-1895)
Similarly, The Cambridge Modern History, volume VII (1903) says:
On December 27, 1861, both the banks and the government by agreement suspended specie payments; and on February 25, 1862, the President signed an Act passed by Congress making non-interest-bearing Treasury notes a legal tender for all debts except duties on imports, and in satisfaction of all claims against the government except for interest upon the public debt, both these exceptions remaining payable in coin.
Is there any proof they were assassinated for this reason?
Regarding the attempted assassination of Jackson:
Lawrence offered a variety of explanations for the shooting. He blamed Jackson for the loss of his job. He claimed that with the President dead, "money would be more plenty," (a reference to Jackson's struggle with the Bank of the United States)
More specifically, according to Thirty Years' View Or a History of the Working of the American (1854), further concerning Lawrence's motivation for attempted assassination:
mechanics would have plenty of work; and that money would be more plenty. On being asked why it would be more plenty, he replied, it would be more easily obtained from the bank. On being asked what bank, he replied the Bank of the United States. On being asked if he knew the president, directors, or any of the officers of the bank, or had ever held any intercourse with them, or knew how he could get money out of the bank, he replied no—that he slightly knew Mr. Smith only.