Javafx resize components when fullscreen
The problem is because you are using AnchorPane as your root pane. Though, you can use AnchorPane for scenarios like this, I personally do not prefer it because you need to do a lot of things to get it right. There are easier ways to do and that is what I am gonna show you.
From the Javadocs :
AnchorPane allows the edges of child nodes to be anchored to an offset from the anchor pane's edges.
Solution
Use a different layout. May be a GridPane or VBox. These layouts have child alignment instead of anchors, which allow the children to be aligned to a particular position.
For re-sizing your children, you can set the HGrow / VGrow property on them.
Example
<?xml version="1.0" encoding="UTF-8"?>
<?import javafx.geometry.*?>
<?import javafx.scene.control.*?>
<?import javafx.scene.layout.*?>
<VBox alignment="CENTER" maxHeight="-Infinity" maxWidth="-Infinity" minHeight="-Infinity" minWidth="-Infinity" prefHeight="400.0" prefWidth="600.0" spacing="20.0" style="-fx-background-color: DARKCYAN;" xmlns="http://javafx.com/javafx/8.0.40" xmlns:fx="http://javafx.com/fxml/1">
<children>
<Label text="Easy Doctor" />
<TextField promptText="Username" />
<TextField promptText="Password" />
</children>
<padding>
<Insets bottom="50.0" left="50.0" right="50.0" top="50.0" />
</padding>
</VBox>
Here is a small gif on how it looks :