Statement I is false because binding energy per nucleon does not increase monotonically with mass number. It peaks at Iron-56 and decreases for heavier elements.
Statement II is true. It describes the fundamental principle of nuclear reactions: fission of heavy nuclei (low BE/A) and fusion of light nuclei (low BE/A) both produce nuclei with higher binding energy per nucleon, releasing energy in the process. Both processes are energetically favorable for the same reason: moving toward higher stability (higher BE/A).