How does "alive tissue" include nails and hair?

The two people who say non-living matter are Kyle, who was told by John, and John, who was told by Sarah, who was told by Kyle... who was told by John. Both John and Kyle are using the Time Displacement Engine for the first time in their respective timelines, and the TDE is the "first tactical time weapon". As we see later on in Genisys, the same Time Displacement Engine we see Kyle sent through, is being built as a prototype in the altered past by the now infected John.

So that means that not only is the information on the TDE is a paradox, but it is also spoken by people in normal conversation, not scientific accuracy.

Instead of thinking it as non-dead, Living Matter can stand to mean Organic Matter, as opposed to inorganic matter. It could also mean Biological Matter. Both would include hair and nails, which while the exposed parts are dead, they are still organic in nature.

The more technical meaning of the Time Displacement Engine's restriction of Living Tissue is that Living Tissue creates a "Bioelectrical Field" which is needed for the TDE to work. Hair and Nails, being bio-compatible with the body, would not block this field, allowing the hair and nail to be included.

Subnote: The clothes rule, based on the Organic Matter rule above, is likely because no one wears 100% organic clothing. The fancy uniforms in the future are likely not made from farm animals or plants. They look synthetic and plastic-y. And even 1973 would be full of synthetic blend fabrics. 1960s had an influx of synthetic fabric due to low cotton crop production.

Hair and fingernails are classified as keratins. "Keratin is the protein that protects epithelial cells from damage or stress that has potential to kill the cell." Keratin is also part of the outer layer of skin and the tongue and hard palate.

While keratin is formed from dead cells, it is clearly a key biological element in the survival of a living organism.

In other words, keratin is a biological element and is different from items such as clothing and it makes sense that it would be included in part of the transported matter since it is an integral part of the organism.