The phrase ‘life unworthy of life‘ is known particularly because of the fact that the Nazis used it when eliminating ‘defectives’ in their Action T4 project (and later, in reference to the Jews), but American and British Eugenicists also were known in terms of ‘lives unworthy of life.’ Quotes are provided below.
Herbert Spencer, CHAPTER XIV.: preparation in biology. The Study of Sociology  [SOURCE]
Other evils, no less serious, are entailed by legislative actions and by actions of individuals, single and combined, which overlook or disregard a kindred biological truth. Besides an habitual neglect of the fact that the quality of a society is physically lowered by the artificial preservation of its feeblest members, there is an habitual neglect of the fact that the quality of a society is lowered morally and intellectually, by the artificial preservation of those who are least able to take care of themselves.
[…]For if the unworthy are helped to increase, by shielding them from that mortality which their unworthiness would naturally entail, the effect is to produce, generation after generation, a greater unworthiness. […]
Fostering the good-for-nothing at the expense of the good, is an extreme cruelty. It is a deliberate storing-up of miseries for future generations. There is no greater curse to posterity than that of bequeathing them an increasing population of imbeciles and idlers and criminals. To aid the bad in multiplying, is, in effect, the same as maliciously providing for our descendants a multitude of enemies.
My purpose is simply to show that a rational policy must recognize certain general truths of Biology; and to insist that only when study of these general truths, as illustrated throughout the living world, has woven them into the conceptions of things, is there gained a strong conviction that disregard of them must cause enormous mischiefs.
W. Duncan McKim, Heredity and Human Progress, 1900. pgs 3-4.
There can be no well-being for an organism but through its nice adaptation to its environment. Such an adaptation we had, doubtless, in the remote past, but we have it no longer. For the lower animals, nature has secured adaptation through selection: the fittest have survived, and these, at each successive stage, have come very near to the perfection then possible. For our earliest ancestors we must believe that nature made similar provision. But as men rose to a higher level of intelligence and united into societies, they began to make for themselves an artificial environment, which grew ever more complex, more many-sided. The valuable principle of cooperation was gradually introduced, but not always applied wisely; and through its misuse the salutary working of natural selection was often, in great measure, set at defiance. When a man, although ill-adapted to the natural environment, presented a fairly adequate adjustment to certain of the many sides of that which was artificial, society often came to his support, paid for him as it were his natural debts, secured his survival, and favored the continuance of his kind. But the sanction thus purchased for many an unworthy life has often proved a curse both for the race and the individual.