Am I entitled to claim compensation for a back injury sustained at work, knowing that my employer does not have insurance, evades paying my dues, and his repeated promises have not been fulfilled?

If the labor law obliges the employer to insure its employees to cover the cost of treatment, and the contract does not stipulate otherwise, then the employer is obligated to bear the expenses of treating the harm that befell you. This is because labor law is considered a custom, and "what is known by custom is like what is stipulated by text." Therefore, you have the right to demand this, unless the contract stipulates that he does not bear the expenses.