For scalar and spinor fields, the one-loop correction to the Bekenstein-Hawking entropy is equal to the entanglement entropy. For gauge fields, the one-loop correction to the Bekenstein-Hawking entropy contains a negative divergent piece (a "contact term") that makes the entropy negative. This contact term has no known statistical interpretation. Most surprisingly, the divergence appears in two dimensions, where gauge fields have no local degrees of freedom. We show that in two dimensions the contact term results from an incorrect treatment of ghost zero modes. In a manifestly gauge-invariant reduced phase space quantization, the gauge field contribution to the Bekenstein-Hawking entropy is positive, finite, and equal to the entanglement entropy.