Random numbers are a fundamental ingredient in fields such as simulation, modeling, and cryptography. Good random numbers should be independent and uniformly distributed. Moreover, for cryptographic applications, they should also be unpredictable. A fundamental feature of quantum theory is that certain measurement outcomes are intrinsically random and unpredictable. These can be harnessed to provide unconditionally secure random numbers. We demonstrate a real-time self-testing source-independent quantum random-number generator (SI QRNG) that uses squeezed light as a source. We generate secure random numbers by measuring the quadratures of the electromagnetic field without making any assumptions about the source other than an energy bound; only the detection device is trusted. We use homodyne detection to measure alternately the (Q) over cap and (P) over cap conjugate quadratures of our source. (P) over cap measurements allow us to estimate a bound on any classical or quantum side information that a malicious eavesdropper may obtain. This bound gives the minimum number of secure bits we can extract from the (Q) over cap measurement. We discuss the performance of different estimators for this bound. We operate this QRNG with a squeezed-state source and compare its performance with a thermal-state source. This is a demonstration of a QRNG using a squeezed state, as well as an implementation of real-time quadrature switching for a SI QRNG.