We systematically study different approaches to introduce high birefringence and high polarization extinction ratio in hollow core antiresonant fibers. Having shown the ineffectiveness of elliptical cores to induce large birefringence in hollow core fibers, we focus on designing and optimizing polarization maintaining Hollow Core Nested Antiresonant Nodeless Fibers (HC-NANF). In a first approach, we create and exploit anti-crossings with glass modes at different wavelengths for the two polarizations. We show that suitable low loss high birefringence regions can be obtained by appropriately modifying the thickness of tubes along one direction while leaving the tubes in the orthogonal direction unchanged and in antiresonance. Using this concept, we propose a new birefringent NANF design providing low loss (~40dB/km) and high birefringence (>10-4) over a record bandwidth of ~550nm, and discuss how bandwidth can be traded off to further reduce the loss to a few dB/km. Finally, we propose a polarization mode-stripping technique in the birefringent NANF. As a demonstration, we propose a polarizing birefringent NANF design that can achieve orthogonal polarization loss ratios as large as 30dB over the C-band while eliminating any undesirable polarization coupling effect thereby resulting in a single polarization output in a hollow core fiber regardless of the input polarization state.