Does an allergic reaction kill body cells?
Is it possible that the human body actually kills its own cells in the
  process?


Yes it is. It is happens not only as a result of allergic reactions (see comment of @terdon). But in case of allergy our body have 3 methods for killing own cells.


Cytotoxic, antibody-dependent (Type 2 hypersensitivity) The antibodies produced by the immune response bind to antigens on the patient's own cell surfaces.These cells are recognized by macrophages or dendritic cells, which act as antigen-presenting cells. This causes a B cell response, wherein antibodies are produced against the foreign antigen.
Immune complex (Type 3 hypersensitivity) Occurs when antigen-antibody complexes that are not adequately cleared by innate immune cells accumulate, giving rise to an inflammatory response and attraction of leukocytes
Cell-mediated immune memory response (Type 4 hypersensitivity) is an immune response that does not involve antibodies, but rather involves the activation of phagocytes, natural killer cells (NK), antigen-specific cytotoxic T-lymphocytes, and the release of various cytokines in response to an antigen.


See Wiki about types and diseases and about Hypersensitivity