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The pairs (U,X) are called Runge pairs. The homology version of your statement is proved in the paper of Andreotti and Narasimhan Annals of Math vol 76 no 3 (1962) 499-509 using Morse Theory.The title of the paper is "A Topological property of Runge pairs" The paper by Coltoiu Mihalache titled On the Homology Groups of Stein spaces and Runge pairs, Journal fur reine und angewandte Mathematik volume 371 no 5 pages 215-220 proves the homology statement for Runge pairs of Stein spaces.