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Hexagonal boron nitride (hBN) has become a mainstay as an insulating barrier in stackable nanoelectronics because of its large bandgap and chemical stability. At mono- and bilayer thicknesses, hBN can function as a tunnel barrier for electronic spectroscopy measurements. Noise spectroscopy is of particular interest, as noise can be a sensitive probe for electronic correlations not detectable by first-moment current measurements. In addition to the expected Johnson-Nyquist thermal noise and nonequilibrium shot noise, low frequency (<100 kHz) noise measurements in Au/hBN/Au tunneling structures as a function of temperature and bias reveal the presence of thermally excited dynamic defects, as manifested through a flicker noise contribution at high bias that freezes out as temperature is decreased. In contrast, broad-band high frequency (∼250MHz – 580MHz) measurements on the same device show shot noise with no flicker noise contribution. The presence of the flicker noise through multiple fabrication approaches and processing treatments suggests that the fluctuators are in the hBN layer itself. Device-to-device variation and the approximate 1/f dependence of the flicker noise constrain the fluctuator density to on the order of a few per square micron.